Questions M1 (2067 questions)

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Edexcel M1 Q5
15 marks Moderate -0.3
\(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors. The point \(A\) has position vector \(6\mathbf{j}\) m relative to an origin \(O\). At time \(t = 0\) a particle \(P\) starts from \(O\) and moves with constant velocity \((5\mathbf{i} + 2\mathbf{j})\) ms\(^{-1}\). At the same instant a particle \(Q\) starts from \(A\) and moves with constant velocity \(4\mathbf{i}\) ms\(^{-1}\).
  1. Write down the position vectors of \(P\) and of \(Q\) at time \(t\) seconds. [3 marks]
  2. Show that the distance \(d\) m between \(P\) and \(Q\) at time \(t\) seconds is such that $$d^2 = 5t^2 - 24t + 36.$$ [5 marks]
  3. Find the value of \(t\) for which \(d^2\) is a minimum. [3 marks]
  4. Hence find the minimum distance between \(P\) and \(Q\), and state the position vector of each particle when they are closest together. [4 marks]
Edexcel M1 Q6
15 marks Standard +0.3
\(A\), \(B\) and \(C\) are three small spheres of equal radii and masses \(2m\), \(m\) and \(5m\) respectively. They are placed in a straight line on a smooth horizontal surface. \(A\) is projected with speed 6 ms\(^{-1}\) towards \(B\), which is at rest. When \(A\) hits \(B\) it exerts an impulse of magnitude \(8m\) Ns on \(B\).
  1. Find the speed with which \(B\) starts to move. [2 marks]
  2. Show that the speed of \(A\) after it collides with \(B\) is 2 ms\(^{-1}\). [3 marks]
After travelling 3 m, \(B\) hits \(C\), which is then travelling towards \(B\) at \(2.2\) ms\(^{-1}\). \(C\) is brought to rest by this impact.
  1. Show that the direction of \(B\)'s motion is reversed and find its new speed. [3 marks]
  2. Find how far \(B\) now travels before it collides with \(A\) again. [6 marks]
  3. State a modelling assumption that you have made about the spheres. [1 mark]
Edexcel M1 Q7
16 marks Standard +0.3
A particle \(P\), of mass \(m\), is in contact with a rough plane inclined at 30° to the horizontal as shown. A light string is attached to \(P\) and makes an angle of 30° with the plane. When the tension in this string has magnitude \(kmg\), \(P\) is just on the point of moving up the plane. \includegraphics{figure_7}
  1. Show that \(\mu\), the coefficient of friction between \(P\) and the plane, is \(\frac{k\sqrt{3} - 1}{\sqrt{3} - k}\). [7 marks]
  2. Given further that \(k = \frac{3\sqrt{3}}{7}\), deduce that \(\mu = \frac{\sqrt{3}}{6}\). [3 marks]
The string is now removed.
  1. Determine whether \(P\) will move down the plane and, if it does, find its acceleration. [5 marks]
  2. Give a reason why the way in which \(P\) is shown in the diagram might not be consistent with the modelling assumptions that have been made. [1 mark]
Edexcel M1 Q1
4 marks Easy -1.8
Briefly define the following terms used in modelling in Mechanics:
  1. lamina,
  2. uniform rod,
  3. smooth surface,
  4. particle.
[4 marks]
Edexcel M1 Q2
8 marks Moderate -0.3
Two forces \(\mathbf{F}\) and \(\mathbf{G}\) are given by \(\mathbf{F} = (6\mathbf{i} - 5\mathbf{j})\) N, \(\mathbf{G} = (3\mathbf{i} + 17\mathbf{j})\) N, where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors in the \(x\) and \(y\) directions respectively and the unit of length on each axis is 1 cm.
  1. Find the magnitude of \(\mathbf{R}\), the resultant of \(\mathbf{F}\) and \(\mathbf{G}\). [3 marks]
  2. Find the angle between the direction of \(\mathbf{R}\) and the positive \(x\)-axis. [2 marks]
\(\mathbf{R}\) acts through the point \(P(-4, 3)\). \(O\) is the origin \((0, 0)\).
  1. Use the fact that \(OP\) is perpendicular to the line of action of \(\mathbf{R}\) to calculate the moment of \(\mathbf{R}\) about an axis through the origin and perpendicular to the \(x\)-\(y\) plane. [3 marks]
Edexcel M1 Q3
12 marks Standard +0.3
A string is attached to a packing case of mass 12 kg, which is at rest on a rough horizontal plane. When a force of magnitude 50 N is applied at the other end of the string, and the string makes an angle of 35° with the vertical as shown, the case is on the point of moving. \includegraphics{figure_3}
  1. Find the coefficient of friction between the case and the plane. [5 marks]
The force is now increased, with the string at the same angle, and the case starts to move along the plane with constant acceleration, reaching a speed of 2 ms\(^{-1}\) after 4 seconds.
  1. Find the magnitude of the new force. [5 marks]
  2. State any modelling assumptions you have made about the case and the string. [2 marks]
Edexcel M1 Q4
12 marks Standard +0.3
A uniform yoke \(AB\), of mass 4 kg and length 4\(a\) m, rests on the shoulders \(S\) and \(T\) of two oxen. \(AS = TB = a\) m. A bucket of mass \(x\) kg is suspended from \(A\). \includegraphics{figure_4}
  1. Show that the vertical force on the yoke at \(T\) has magnitude \((2 - \frac{1}{4}x)g\) N and find, in terms of \(x\) and \(g\), the vertical force on the yoke at \(S\). [7 marks]
  2. If the ratio of these vertical forces is \(5 : 1\), find the value of \(x\). [3 marks]
  3. Find the maximum value of \(x\) for which the yoke will remain horizontal. [2 marks]
Edexcel M1 Q5
12 marks Standard +0.3
Two small smooth spheres \(A\) and \(B\), of equal radius but masses \(m\) kg and \(km\) kg respectively, where \(k > 1\), move towards each other along a straight line and collide directly. Immediately before the collision, \(A\) has speed 5 ms\(^{-1}\) and \(B\) has speed 3 ms\(^{-1}\). In the collision, the impulse exerted by \(A\) on \(B\) has magnitude \(7km\) Ns.
  1. Find the speed of \(B\) after the impact. [3 marks]
  2. Show that the speed of \(A\) immediately after the collision is \((7k - 5)\) ms\(^{-1}\) and deduce that the direction of \(A\)'s motion is reversed. [5 marks]
\(B\) is now given a further impulse of magnitude \(mu\) Ns, as a result of which a second collision between it and \(A\) occurs.
  1. Show that \(u > k(7k - 1)\). [4 marks]
Edexcel M1 Q6
13 marks Moderate -0.3
The velocity-time graph illustrates the motion of a particle which accelerates from rest to 8 ms\(^{-1}\) in \(x\) seconds and then to 24 ms\(^{-1}\) in a further 4 seconds. It then travels at a constant speed for another \(y\) seconds before decelerating to 12 ms\(^{-1}\) over the next \(y\) seconds and then to rest in the final 7 seconds of its motion. \includegraphics{figure_6} Given that the total distance travelled by the particle is 496 m,
  1. show that \(2x + 21y = 195\). [4 marks]
Given also that the average speed of the particle during its motion is 15.5 ms\(^{-1}\),
  1. show that \(x + 2y = 21\). [3 marks]
  2. Hence find the values of \(x\) and \(y\). [3 marks]
  3. Write down the acceleration for each section of the motion. [3 marks]
Edexcel M1 Q7
14 marks Standard +0.8
Two particles \(P\) and \(Q\), of masses \(2m\) and \(3m\) respectively, are connected by a light string. Initially, \(P\) is at rest on a smooth horizontal table. The string passes over a small smooth pulley and \(Q\) rests on a rough plane inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac{4}{3}\). The coefficient of friction between \(Q\) and the inclined plane is \(\frac{1}{6}\). \includegraphics{figure_7} The system is released from rest with \(Q\) at a distance of 0.8 metres above a horizontal floor.
  1. Show that the acceleration of \(P\) and \(Q\) is \(\frac{21g}{50}\), stating a modelling assumption which you must make to ensure that both particles have the same acceleration. [7 marks]
  2. Find the speed with which \(Q\) hits the floor. [2 marks]
After \(Q\) hits the floor and does not rebound, \(P\) moves a further 0.2 m until it hits the pulley.
  1. Find the total time after the system is released before \(P\) hits the pulley. [5 marks]
Edexcel M1 Q1
7 marks Moderate -0.3
A boy holds a 30 cm metal ruler between three fingers of one hand, pushing down with the middle finger and up with the other two, at the points marked 5 cm, 10 cm and \(x\) cm and exerting forces of magnitude 11 N, 18 N and 8 N respectively. The ruler is in equilibrium in this position. Modelling the ruler as a uniform rod, find \includegraphics{figure_1}
  1. the mass of the ruler, in grams, \hfill [3 marks]
  2. the value of \(x\). \hfill [3 marks]
  3. State how you have used the modelling assumption that the ruler is a uniform rod. \hfill [1 mark]
Edexcel M1 Q2
7 marks Standard +0.8
\includegraphics{figure_2} A small packet of mass 0.3 kg rests on a rough horizontal surface. The coefficient of friction between the packet and the surface is \(\frac{1}{4}\). Two strings are attached to the packet, making angles of 45° and 30° with the horizontal, and when forces of magnitude 2 N and \(F\) N are exerted through the strings as shown, the packet is on the point of moving in the direction \(\overrightarrow{AB}\). Find the value of \(F\). \hfill [7 marks]
Edexcel M1 Q3
7 marks Moderate -0.8
A body moves in a straight line with constant acceleration. Its speed increases from 17 ms\(^{-1}\) to 33 ms\(^{-1}\) while it travels a distance of 250 m. Find
  1. the time taken to travel the 250 m, \hfill [3 marks]
  2. the acceleration of the body. \hfill [2 marks]
The body now decelerates at a constant rate from 33 ms\(^{-1}\) to rest in 6 seconds.
  1. Find the distance travelled in these 6 seconds. \hfill [2 marks]
Edexcel M1 Q4
12 marks Standard +0.3
A particle \(P\) of mass \(m\) kg, at rest on a smooth horizontal table, is connected to particles \(Q\) and \(R\), of mass 0.1 kg and 0.5 kg respectively, by strings which pass over fixed pulleys at the edges of the table. The system is released from rest with \(Q\) and \(R\) hanging freely and it is found that the tension in the section of the string between \(P\) and \(R\) is 2 N.
  1. Show that the acceleration of the particles has magnitude 5.8 ms\(^{-2}\). \hfill [3 marks]
  2. Find the value of \(m\). \hfill [5 marks]
Modelling assumptions have been made about the pulley and the strings.
  1. Briefly describe these two assumptions. For each one, state how the mathematical model would be altered if the assumption were not made. \hfill [4 marks]
Edexcel M1 Q5
12 marks Standard +0.3
Two trucks \(P\) and \(Q\), of masses 18 000 kg and 16 000 kg respectively, collide while moving towards each other in a straight line. Immediately before the collision, both trucks are travelling at the same speed, \(u\) ms\(^{-1}\). Immediately after the collision, \(P\) is moving at half its original speed, its direction of motion having been reversed.
  1. Find, in terms of \(u\), the speed of \(Q\) immediately after the collision. \hfill [5 marks]
  2. State, with a reason, whether the direction of \(Q\)'s motion has been reversed. \hfill [1 mark]
  3. Find, in terms of \(u\), the magnitude of the impulse exerted by \(P\) on \(Q\) in the collision, stating the units of your answer. \hfill [3 marks]
The force exerted by each truck on the other in the impact has magnitude \(108000u\) N.
  1. Find the time for which the trucks are in contact. \hfill [3 marks]
Edexcel M1 Q6
13 marks Moderate -0.8
A particle \(P\) moves in a straight line such that its displacement from a fixed point \(O\) at time \(t\) s is \(y\) metres. The graph of \(y\) against \(t\) is as shown.
[diagram]
  1. Write down the velocity of \(P\) when
    1. \(t = 1\), \quad (ii) \(t = 10\). \hfill [2 marks]
  2. State the total distance travelled by \(P\). \hfill [2 marks]
  3. Write down a formula for \(y\) in terms of \(t\) when \(2 \leq t < 4\). \hfill [3 marks]
  4. Sketch a velocity-time graph for the motion of \(P\) during the twelve seconds. \hfill [3 marks]
  5. Find the maximum speed of \(P\) during the motion. \hfill [3 marks]
Edexcel M1 Q7
17 marks Standard +0.3
Two trains \(S\) and \(T\) are moving with constant speeds on straight tracks which intersect at the point \(O\). At 9.00 a.m. \(S\) has position vector \((-10\mathbf{i} + 24\mathbf{j})\) km and \(T\) has position vector \(25\mathbf{j}\) km relative to \(O\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors in the directions due east and due north respectively. \(S\) is moving with speed 52 km h\(^{-1}\) and \(T\) is moving with speed 50 km h\(^{-1}\), both towards \(O\).
  1. Show that the velocity vector of \(S\) is \((20\mathbf{i} - 48\mathbf{j})\) km h\(^{-1}\) and find the velocity vector of \(T\). \hfill [5 marks]
  2. Find expressions for the position vectors of \(S\) and \(T\) at time \(t\) minutes after 9.00 a.m. \hfill [5 marks]
  3. Show that the bearing of \(T\) from \(S\) remains constant during the motion, and find this bearing. \hfill [5 marks]
  4. Show that if the trains continue at the given speeds they will collide. \hfill [2 marks]
OCR M1 Q1
7 marks Moderate -0.3
\includegraphics{figure_1} A light inextensible string has its ends attached to two fixed points \(A\) and \(B\). The point \(A\) is vertically above \(B\). A smooth ring \(R\) of mass \(m\) kg is threaded on the string and is pulled by a force of magnitude \(1.6\) N acting upwards at \(45°\) to the horizontal. The section \(AR\) of the string makes an angle of \(30°\) with the downward vertical and the section \(BR\) is horizontal (see diagram). The ring is in equilibrium with the string taut.
  1. Give a reason why the tension in the part \(AR\) of the string is the same as that in the part \(BR\). [1]
  2. Show that the tension in the string is \(0.754\) N, correct to 3 significant figures. [3]
  3. Find the value of \(m\). [3]
OCR M1 Q2
7 marks Standard +0.3
\includegraphics{figure_2} Particles \(A\) and \(B\), of masses \(0.2\) kg and \(0.3\) kg respectively, are attached to the ends of a light inextensible string. Particle \(A\) is held at rest at a fixed point and \(B\) hangs vertically below \(A\). Particle \(A\) is now released. As the particles fall the air resistance acting on \(A\) is \(0.4\) N and the air resistance acting on \(B\) is \(0.25\) N (see diagram). The downward acceleration of each of the particles is \(a\) m s\(^{-2}\) and the tension in the string is \(T\) N.
  1. Write down two equations in \(a\) and \(T\) obtained by applying Newton's second law to \(A\) and to \(B\). [4]
  2. Find the values of \(a\) and \(T\). [3]
OCR M1 Q3
8 marks Standard +0.3
Two small spheres \(P\) and \(Q\) have masses \(0.1\) kg and \(0.2\) kg respectively. The spheres are moving directly towards each other on a horizontal plane and collide. Immediately before the collision \(P\) has speed \(4\) m s\(^{-1}\) and \(Q\) has speed \(3\) m s\(^{-1}\). Immediately after the collision the spheres move away from each other, \(P\) with speed \(u\) m s\(^{-1}\) and \(Q\) with speed \((3.5 - u)\) m s\(^{-1}\).
  1. Find the value of \(u\). [4]
After the collision the spheres both move with deceleration of magnitude \(5\) m s\(^{-2}\) until they come to rest on the plane.
  1. Find the distance \(PQ\) when both \(P\) and \(Q\) are at rest. [4]
OCR M1 Q4
9 marks Standard +0.3
A particle moves downwards on a smooth plane inclined at an angle \(\alpha\) to the horizontal. The particle passes through the point \(P\) with speed \(u\) m s\(^{-1}\). The particle travels \(2\) m during the first \(0.8\) s after passing through \(P\), then a further \(6\) m in the next \(1.2\) s. Find
  1. the value of \(u\) and the acceleration of the particle, [7]
  2. the value of \(\alpha\) in degrees. [2]
OCR M1 Q5
12 marks Standard +0.8
\includegraphics{figure_5} Two small rings \(A\) and \(B\) are attached to opposite ends of a light inextensible string. The rings are threaded on a rough wire which is fixed vertically. \(A\) is above \(B\). A horizontal force is applied to a point \(P\) of the string. Both parts \(AP\) and \(BP\) of the string are taut. The system is in equilibrium with angle \(BAP = \alpha\) and angle \(ABP = \beta\) (see diagram). The weight of \(A\) is \(2\) N and the tensions in the parts \(AP\) and \(BP\) of the string are \(7\) N and \(T\) N respectively. It is given that \(\cos \alpha = 0.28\) and \(\sin \alpha = 0.96\), and that \(A\) is in limiting equilibrium.
  1. Find the coefficient of friction between the wire and the ring \(A\). [7]
  2. By considering the forces acting at \(P\), show that \(T \cos \beta = 1.96\). [2]
  3. Given that there is no frictional force acting on \(B\), find the mass of \(B\). [3]
OCR M1 Q6
12 marks Standard +0.3
A particle of mass \(0.04\) kg is acted on by a force of magnitude \(P\) N in a direction at an angle \(\alpha\) to the upward vertical.
  1. The resultant of the weight of the particle and the force applied to the particle acts horizontally. Given that \(\alpha = 20°\) find
    1. the value of \(P\), [3]
    2. the magnitude of the resultant, [2]
    3. the magnitude of the acceleration of the particle. [2]
  2. It is given instead that \(P = 0.08\) and \(\alpha = 90°\). Find the magnitude and direction of the resultant force on the particle. [5]
OCR M1 Q7
17 marks Standard +0.3
\includegraphics{figure_7} A car \(P\) starts from rest and travels along a straight road for \(600\) s. The \((t, v)\) graph for the journey is shown in the diagram. This graph consists of three straight line segments. Find
  1. the distance travelled by \(P\), [3]
  2. the deceleration of \(P\) during the interval \(500 < t < 600\). [2]
Another car \(Q\) starts from rest at the same instant as \(P\) and travels in the same direction along the same road for \(600\) s. At time \(t\) s after starting the velocity of \(Q\) is \((600t^2 - t^3) \times 10^{-6}\) m s\(^{-1}\).
  1. Find an expression in terms of \(t\) for the acceleration of \(Q\). [2]
  2. Find how much less \(Q\)'s deceleration is than \(P\)'s when \(t = 550\). [2]
  3. Show that \(Q\) has its maximum velocity when \(t = 400\). [2]
  4. Find how much further \(Q\) has travelled than \(P\) when \(t = 400\). [6]
OCR M1 Q1
7 marks Standard +0.2
\includegraphics{figure_1} Particles \(P\) and \(Q\), of masses \(0.3\) kg and \(0.4\) kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The system is in motion with the string taut and with each of the particles moving vertically. The downward acceleration of \(P\) is \(a\) m s\(^{-2}\) (see diagram).
  1. Show that \(a = -1.4\). [4]
Initially \(P\) and \(Q\) are at the same horizontal level. \(P\)'s initial velocity is vertically downwards and has magnitude \(2.8\) m s\(^{-1}\).
  1. Assuming that \(P\) does not reach the floor and that \(Q\) does not reach the pulley, find the time taken for \(P\) to return to its initial position. [3]